Part (i).
For every fixed \( h > 0 \) , \( \widehat{m}^h \)  is in \( \mathcal {C}([0,T]; \mathbb {L}^2_w \cap \mathring{\mathbb {H}}^1) \)  and so does \( \overline{m}^h \) . Then part (i) follows directly from the estimates in (REF ) and (REF ).
Part (ii). Recall (REF ), we have from the definition (REF ) that
\(\overline{m}^h(t,x)&= I_0(x) + I_1(t,x) + I_2(t,x) + I_3(t,x) + I_4(t,x),\) 

where for \( x \in [x_k,x_{k+1}) \) , \( k \in \mathbb {Z}\) ,
\(I_0(x)&= \frac{1}{2} \left(m_0(x_k) + m_0(x_{k-1}) \right)+ (x-x_k) \partial ^{h} m_0(x_{k-1}) + \frac{1}{2}(x-x_k)^2\Delta ^h m_0(x_k), \\I_1(t,x)&= \frac{1}{2} \int _0^t \left(F^h(\widehat{m}^h(s,x_k)) + \frac{1}{2} S^h(\widehat{m}^h(s,x_k)) \right)ds \\&\quad + \frac{1}{2} \int _0^t \left(F^h(\widehat{m}^h(s,x_{k-1})) + \frac{1}{2} S^h(\widehat{m}^h(s,x_{k-1})) \right)ds, \\I_2(t,x)&= (x-x_k) \int _0^t \partial ^{h} \left(F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right)(s,x_{k-1}) \ ds, \\I_3(t,x)&= \frac{1}{2}(x-x_k)^2 \int _0^t \Delta ^h \left(F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right)(s,x_k) \ ds, \\I_4(t,x)&= \frac{1}{2} \int _0^t \left(G^h(\widehat{m}^h(s,x_k)) + G^h(\widehat{m}^h(s,x_{k-1})) \right)d\widehat{W}^h(s) \\&\quad + (x-x_k) \int _0^t \partial ^{h} G^h(\widehat{m}^h(s,x_{k-1})) \ d\widehat{W}^h(s) \\&\quad + \frac{1}{2}(x-x_k)^2 \int _0^t \Delta ^h G^h(\widehat{m}^h(s,x_k)) \ d\widehat{W}^h(s).\) 

By the \( \mathbb {L}^\infty \) -estimate of \( \overline{m}^h \)  in (REF ),
\(\mathbb {E}\left[ \int _0^T |\overline{m}^h(t)|_{\mathbb {L}^2_w}^2 \ dt \right] \le 5^2 \pi T.\) 

For \( I_1 \) , by Lemma REF , there exists a constant \( a_1 \)  that may depend on \( C_v, C_\kappa , \alpha , \gamma , T, K_{1,1}, K_{1,3} \)  and \( K_{2,1} \)  such that
\(\mathbb {E}\left[ \int _0^T \left| F^h(\widehat{m}^h(t)) + \frac{1}{2} S^h(\widehat{m}^h(t)) \right|_{\mathbb {L}^2}^2 \ dt \right]\le a_1,\) 

For \( I_2 \)  and \( I_3 \) , since \( |x-x_k| \le h \)  and
\(h| \partial ^{h} u(x_{k-1}) | &\le |u(x_k)| + |u(x_{k-1})|, \\h^2 |\Delta ^h u(x_k) | &\le |u(x_{k+1})| + |u(x_{k-1})| + 2|u(x_k)|,\) 

there also exist constants \( a_2, a_3 \)  such that
\(\mathbb {E}\left[ \int _0^T \left| h \partial ^{h} \left(F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right)(t) \right|_{\mathbb {L}^2}^2 dt \right]&\le a_2, \\\mathbb {E}\left[ \int _0^T \left| h^2 \Delta ^h \left(F^h(\widehat{m}^h) + \frac{1}{2} S^h(\widehat{m}^h) \right)(t) \right|_{\mathbb {L}^2}^2 dt \right]&\le a_3.\) 
Similarly, for the stochastic integrals in \( I_4 \) , we only need to verify that \( \int _0^T G^h(\widehat{m}^h(s)) \ d\widehat{W}^h(s) \)  is bounded in \( L^p(\Omega ; W^{\alpha , p}(0,T; \mathbb {L}^2)) \) .
By [1]}, there exist a constant \( C \)  depending on \( \alpha , p, \gamma , T \)  and a constant \( a_4 \)  depending on \( C, C_\kappa \)  and \( K_{1,p} \)  such that for \( p \in [2,\infty ) \)  and \( \alpha \in (0, \frac{1}{2}) \) ,
\(\mathbb {E}\left[ \left| \int _0^T G^h(\widehat{m}^h(s)) \ d\widehat{W}^h(s) \right|^p_{W^{\alpha ,p}(0,T; \mathbb {L}^2)} \right]&= \mathbb {E}\left[ \left| \int _0^T G^h(m^h(s)) \ dW(s) \right|^p_{W^{\alpha ,p}(0,T; \mathbb {L}_h^2)} \right] \\&\le C \ \mathbb {E}\left[ \int _0^T \left(\sum _j q_j^2 \left| f_j G^h(m^h(s)) \right|^2_{\mathbb {L}_h^2} \right)^\frac{p}{2} ds \right] \\&\le C |\kappa |_{\mathbb {L}^\infty }^p \mathbb {E}\left[\int _0^T \left|G^h(\widehat{m}^h(s)) \right|^p_{\mathbb {L}^2} \ ds \right]\le a_4.\) 
 Since \( \mathbb {L}^2 \hookrightarrow \mathbb {L}^2_w \)  for \( w \ge 1 \) , the estimates above hold for the \( \mathbb {L}^2_w \) -norm.
By Lemma REF , the embedding \( W^{1,2}(0,T; \mathbb {L}^2_w) \hookrightarrow W^{\alpha ,p}(0,T; \mathbb {L}^2_w) \)  is continuous for \( \alpha - \frac{1}{p} < \frac{1}{2} \) .
Thus,
\(\sup _h \mathbb {E}\left[ |\overline{m}^h|_{W^{\alpha ,p}(0,T; \mathbb {L}^2_w)}^2 \right] < \infty .\) 
Part (iii). Since \( |m^h(t,x)|=1 \) , \( \mathbb {P}\) -a.s. for all \( (t,x) \in [0,T] \times \mathbb {Z}_h \) , we observe that
\(|\overline{m}^h(t,x)| \le 1 + \left| \partial ^h m^h(t,x_{k-1})(x-x_k) + \frac{1}{2} \Delta ^h m^h(t,x_k)(x-x_k)^2 \right|,\) 

for \( (t,x) \in [0,T] \times [x_k,x_{k+1}) \) , \( k \in \mathbb {Z}\) . This implies
\(&\mathbb {E}\left[ \int _0^T \int _\mathbb {R}\left| |\overline{m}^h(t,x)| -1 \right|^2 \ dx \ dt \right] \\&\le \mathbb {E}\left[ \int _0^T \sum _k \int _{x_k}^{x_{k+1}} \left| \partial ^{h} m^h(t,x_{k-1})(x-x_k) + \frac{1}{2} \Delta ^h m^h(t,x_k)(x-x_k)^2 \right|^2 \ dx \ dt \right] \\&\le \frac{2}{3} h^2 K_{1,1} T + \frac{1}{10} h^4 K_{3,1},\) 

where the last inequality holds by Lemmata REF  and REF , and we obtain the convergence after taking \( h \rightarrow 0 \) .
